Recursive confidence band construction for an unknown distribution function

Given a sample X1,...,Xn of independent observations from an unknown continuous distribution function F, the problem of constructing a confidence band for F is considered, which is a fundamental problem in statistical inference. This confidence band provides simultaneous inferences on all quantiles...

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Bibliographic Details
Published in:Biometrical journal 2015-01, Vol.57 (1), p.39-51
Main Authors: Kiatsupaibul, Seksan, Hayter, Anthony J.
Format: Article
Language:English
Subjects:
Asymptotic methods
Confidence
Confidence bands
Confidence Intervals
Multinomial distribution
Probability
Quantiles
Recursive methodologies
Simultaneous confidence intervals
Statistical analysis
Statistics as Topic - methods
Uncertainty
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Summary:Given a sample X1,...,Xn of independent observations from an unknown continuous distribution function F, the problem of constructing a confidence band for F is considered, which is a fundamental problem in statistical inference. This confidence band provides simultaneous inferences on all quantiles and also on all of the cumulative probabilities of the distribution, and so they are among the most important inference procedures that address the issue of multiplicity. A fully nonparametric approach is taken where no assumptions are made about the distribution function F. Historical approaches to this problem, such as Kolmogorov's famous () procedure, represent some of the earliest inference methodologies that address the issue of multiplicity. This is because a confidence band at a given confidence level 1−α allows inferences on all of the quantiles of the distribution, and also on all of the cumulative probabilities, at that specified confidence level. In this paper it is shown how recursive methodologies can be employed to construct both one‐sided and two‐sided confidence bands of various types. The first approach operates by putting bounds on the cumulative probabilities at the data points, and a recursive integration approach is described. The second approach operates by providing bounds on certain specified quantiles of the distribution, and its implementation using recursive summations of multinomial probabilities is described. These recursive methodologies are illustrated with examples, and R code is available for their implementation.
ISSN:0323-3847
1521-4036
DOI:10.1002/bimj.201300213